[Proj] Troubles with Newton-Raphson inverse projections

[Gerald I. Evenden]

I finished the basics of the Newton-Raphson general inverse projection method 
(as described in the Turkish paper) about a week ago.

One additional problem with the Turkish paper is that they understated the 
problem of making an appropriate initial estimation of the lon/lat solution.  
If the estimate is sufficiently poor, the looping process can follow the 
wrong path to a solution or simply fail to converge.  Also, the poorer the 
estimation, more loops are required to converge to a solution.  At the moment 
I am pursuing developing a low degree polynomial estimation function which 
will hopefully improve the initial estimate process. 

Secondly, finding the root may also be difficult when the precision of the 
derivatives start to fail.  This is the case near the poles of both the 
Hammer and Aitoff projections.  At a latitude greater than 89° the method 
fails for both these projections.  One also gets suspicious that this will 
occur when the nature of the curve flattens out like the top of a sine curve.  
I have not tried the flat pole Winkel Triplel yet.

This is just an alert to the readership that the Turkish method *is not* a 
cureall for determining the inverse projection.

-- 
The whole religious complexion of the modern world is due
to the absence from Jerusalem of a lunatic asylum.
-- Havelock Ellis (1859-1939) British psychologist